On the fractional Yamabe problem with isolated singularities and related issues

Premi Extraordinari de Doctorat, promoció 2016-2017. Àmbit de CiènciesMy research is based on non-local elliptic semilinear equations in conformal geometry. The fractional curvature is defined from the conformal fractional Laplacian and it is a non-local version of some of the classical local curvatures such as the scalar curvature, the fourth-order Q-curvature or the mean curvature. This new notion of non-local curvature has good conformal properties that allow to treat classical problems from a more general convexity point of view. Note that the fractional curvature in my research is different from the one defined by Caffarelli, Roquejoffre and Savin . In particular, I have worked on the fractional singular Yamabe problem and related issues. This problem arises in conformal geometry when we try to find a conformal metric to a given one having constant fractional curvature and prescribed singularities. The precise problem I considered in my thesis was to find solutions for the fractional Yamabe problem in the Eucliden space of dimension bigger than 2¿, 0<¿<1, with prescribed isolated singularities: first, I just considered radial solutions when there is an isolated singularity and, later, the problem of removing a finite number of points. This thesis consists of nine chapters. First, we give is a brief introduction and summary of the thesis. Next, provide some background, notation and known results. Later, we show the main results, i.e, Chapters 3, 4, 5 and 6. After this, we introduce the research plan to come. The thesis also has two appendixes with useful computations. I started my research focusing on the geometric interpretation of the problem for an isolated singularity (Chapter 3). This study is based on an extension problem for the computation of the conformal fractional Laplacian. This is a Dirichlet-to-Neumann problem for a degenerate elliptic, but local, equation, which gives an example of a boundary reaction problem where the nonlinearity is of power type with the critical Sobolev exponent. Later, I treated the problem as an integro-differential equation, facing two main difficulties: the lack of compactness and the fact that we are dealing with a non-local ODE (Chapter 4) . Our study is carried out using variational methods and it proves the existence of Delaunay-type solutions for the problem. These are radially symmetric metrics with constant fractional curvature. Finally, I applied some gluing methods together with a Lyapunov reduction to construct solutions for the singular fractional Yamabe problem when the singular set consists of a given finite number of points (Chapter 5). At the moment, I am working on the fractional Caffarelli-Kohn-Nirenberg inequality, which is an interpolation between the Hardy and Sobolev fractional inequalities. In particular, I am looking at the radial symmetry or symmetry breaking of the minimizers (Chapter 6). I have collaborated with Weiwei Ao (University of British Columbia), María del Mar González (Universidad Autónoma de Madrid), Manuel del Pino (Universidad de Santiago, Chile) and Juncheng Wei (University of British Columbia).Mi investigación se basa en ecuaciones no-locales elípticas semilineales que surgen en la geometría conforme. La curvatura fraccionaria se define partir del Laplaciano fraccionario conforme y es una versión no local de algunas de las curvaturas locales clásicas tales como la curvatura escalar, la Q-curvatura (operador de orden 4) o la curvatura media. Esta nueva noción de curvatura no-local tiene buenas propiedades conformes que permiten tratar problemas clásicos desde un punto de vista de convexidad más general. Debemos tener en cuenta que se trata de una curvatura fraccionaria distinta a la definida por Caffarelli, Roquejoffre y Savin . En particular, he trabajado en el problema de Yamabe singular fraccionario y problemas relacionados. Se trata de un problema que aparece en la geometría conforme cuando intentamos encontrar una métrica conforme a la dada en una variedad, que tenga curvatura fraccionaria constante y singularidades prescritas. Más concretamente, el problema que he considerado para mi tesis es encontrar soluciones para el problema de Yamabe fraccionario en el espacio Euclídeo de dimensión mayor que 2¿, 0<¿